Optimal. Leaf size=18 \[ 3 (4+x)+\log \left (25+9 \left (e^2+x\right )^2\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {1657, 628} \begin {gather*} \log \left (9 x^2+18 e^2 x+9 e^4+25\right )+3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1657
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3+\frac {18 \left (e^2+x\right )}{25+9 e^4+18 e^2 x+9 x^2}\right ) \, dx\\ &=3 x+18 \int \frac {e^2+x}{25+9 e^4+18 e^2 x+9 x^2} \, dx\\ &=3 x+\log \left (25+9 e^4+18 e^2 x+9 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.50 \begin {gather*} 3 \left (x+\frac {1}{3} \log \left (25+9 e^4+18 e^2 x+9 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 21, normalized size = 1.17 \begin {gather*} 3 \, x + \log \left (9 \, x^{2} + 18 \, x e^{2} + 9 \, e^{4} + 25\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 21, normalized size = 1.17 \begin {gather*} 3 \, x + \log \left (9 \, x^{2} + 18 \, x e^{2} + 9 \, e^{4} + 25\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 22, normalized size = 1.22
method | result | size |
default | \(3 x +\ln \left (9 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +9 x^{2}+25\right )\) | \(22\) |
risch | \(3 x +\ln \left (9 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +9 x^{2}+25\right )\) | \(22\) |
norman | \(3 x +\ln \left (9 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +9 x^{2}+25\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 21, normalized size = 1.17 \begin {gather*} 3 \, x + \log \left (9 \, x^{2} + 18 \, x e^{2} + 9 \, e^{4} + 25\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 17, normalized size = 0.94 \begin {gather*} 3\,x+\ln \left (x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4+\frac {25}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 22, normalized size = 1.22 \begin {gather*} 3 x + \log {\left (9 x^{2} + 18 x e^{2} + 25 + 9 e^{4} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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